Optimal portfolio liquidation in target zone
models and catalytic superprocesses
Eyal Neuman
Imperial College London
Joint work with Alexander SchiedSeptember 29, 2016

Outline
The Control Problem
Financial Motivation
What is a Catalytic-Superprocesses ?
Unique Solution to the Control Problem

Optimal portfolio liquidation in target zone models

The Control Process
1. S = {S(t)}t≥0 is a diffusion process with S(0) = z, reflected atsome barrier c ∈ R.
2. L = {Lt }t≥0 is the local time of S at c.

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Optimal portfolio liquidation in target zone models

The Control Process
1. S = {S(t)}t≥0 is a diffusion process with S(0) = z, reflected atsome barrier c ∈ R.
2. L = {Lt }t≥0 is the local time of S at some c ∈ R.
3. Let X denote the class of all progressively measurable control
R
processes ξ for which 0T |ξt | dLt < ∞ Pz -a.s. for all T > 0 andz ∈ R.

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Optimal portfolio liquidation in target zone models

4. For ξ ∈ X and x0 ∈ R we defineXtξ := x0 +Eyal Neuman | Imperial College London

Z t

ξs dLs , t ≥ 0.

0

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Optimal portfolio liquidation in target zone models

The Control Problem
We consider the minimization of the cost functional for some p ≥ 2,

Z T

Ez

|ξt |p L(dt) +

0

Z T
0

φ(St )|Xtξ |p dt + %(ST )|XTξ |p

1. For ξ ∈ X and
Xtξ := x0 +Eyal Neuman | Imperial College London

Z t

ξs dLs , t ≥ 0.

0

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Optimal portfolio liquidation in target zone models

The Control Problem
We consider the minimization of the cost functional for some p ≥ 2,

Z T

Ez
0

p

|ξt | L(dt) +Z T
0φ(St )|Xtξ |p dt + %(ST )|XTξ |p

1. φ is a bounded measurable function.
2. % ≥ 0 is a bounded continuous penalty function.

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Optimal portfolio liquidation in target zone models

Financial Motivation
Reflected Processes and Target Zone Models

1. Reflecting diffusion processes often be used in models for
currency exchange rates in a target zone.

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Optimal portfolio liquidation in target zone models

Financial Motivation: Reflected Processes andTarget Zone Models
1. Reflecting diffusion processes often be used in models for
currency exchange rates in a target zone.
2. A target zone refers to a regime in which the exchange
rate of a currency is kept within a certain range ofvalues, either through an international agreement or
through central bank intervention.

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Optimal portfolio liquidation in target zone models

Financial Motivation: Reflected Processes andTarget Zone Models
2. A target zone refers to a regime in which the exchange rateof a currency is kept within a certain range of values, either
through an international agreement or through central bankintervention.
3. See for example: Krugman (1991), Svensson (1991),
Bertolla (1991), Bertolla and Caballero (1992), De
Jong (1994), and Ball and Roma (1998).

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Optimal portfolio liquidation in target zone models

Target zone models: HKD/USD

Plot of the HKD/USD exchange rate from 2007 until 2016(currencyconverter.io).

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Optimal portfolio liquidation in target zone models

Target zone models: EUR/CHF
1.24
1.22
1.20
1.18
1.16
1.14
1.12

22 Oct 12

4 Jun 1216 Jan 12

29 Aug 11

1.10

Plot of the EUR/CHF exchange rate from September 1, 2011 throughDecember 31, 2012.
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Optimal portfolio liquidation in target zone models

Target zone models: reflected geometric Brownian motion
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.2

0.4

0.6

0.8

1.0

Plot of reflected geometric Brownian motion reflected at c=1.2.

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Optimal portfolio liquidation in target zone models

Financial Motivation: The Control Process
Xtξ = x0 + 0t ξs dLs , 0 ≤ t ≤ T.
R

1. The control ξ will be interpreted as a trading strategy that
executes orders at infinitesimal rate ξt dLt at those times t
at which St = c.

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Optimal portfolio liquidation in target zone models

Financial Motivation: The Control Process
Xtξ = x0 + 0t ξs dLs , 0 ≤ t ≤ T.
R

1. The control ξ will be interpreted as a trading strategy that
executes orders at infinitesimal rate ξt dLt at those times t
at which St = c.
2. For instance, for an investor wishing to sell Swiss francs
during the period of a lower bound on the EUR/CHF
exchange rate.

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Optimal portfolio liquidation in target zone models

Financial Motivation: The Control Process
1. The control ξ will be interpreted as a trading strategy that
executes orders at infinitesimal rate ξt dLt at those times t
at which St = c.
2. For instance, for an investor wishing to sell Swiss francs during
the period of a lower bound on the EUR/CHF exchange rate.
3. The resulting process Xtξ = x0 + 0t ξs dLs describes theinventory of the investor at time t.
R

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Optimal portfolio liquidation in target zone models

Financial Motivation: The Control Problem
Recall that we wish to minimize of the cost functional for some p ≥ 2,

Z T

Ez

|ξt |p L(dt) +

0

Z T
0

φ(St )|Xtξ |p dt + %(ST )|XTξ |p

1. The expectation of 0T φ(St )|Xtξ |p dt can be regarded as a measurefor the risk associated with holding the position Xtξ at time t.
R

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Optimal portfolio liquidation in target zone models

Financial Motivation: The Control Problem
Recall that we wish to minimize of the cost functional for some p ≥ 2,

Z T

Ez

|ξt |p L(dt) +

0

Z T
0

φ(St )|Xtξ |p dt + %(ST )|XTξ |p

1. The expectation of 0T φ(St )|Xtξ |p dt can be regarded as a measurefor the risk associated with holding the position Xtξ at time t.
R

2. See [Almgren 2012, Forsyth et al 2012, Tse et al 2013, Schied 2013].

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Optimal portfolio liquidation in target zone models

Financial Motivation: The Control Problem
Recall that we wish to minimize of the cost functional for some p ≥ 2,

Z T

Ez
0

|ξt |p L(dt) +Z T
0φ(St )|Xtξ |p dt + %(ST )|XTξ |p

3. Similarly, the expectation of the term %(ST )|XTξ |p can be viewed asa penalty for still keeping the position XTξ at the end of the trading
horizon.

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Optimal portfolio liquidation in target zone models

Financial Motivation: The Control Problem
Recall that we wish to minimize of the cost functional for some p ≥ 2,

Z T

Ez
0

|ξt |p L(dt) +Z T
0φ(St )|Xtξ |p dt + %(ST )|XTξ |p

4. The term 0T |ξt |p L(dt) can be interpreted as a cost term that arises
from the temporary price impact generated by executing the strategy
Xξ.
R

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Optimal portfolio liquidation in target zone models

Temporary Impact Costs Caused by {Xtξ }t≥0
We focus in the case where {St }t≥0 is a Brownian motion with drift.
For n ∈ N fixed, we define the following stopping times
(n)

τ0
(n)

τk

:= inf t ≥ 0 St ∈ c + 2−n Z ,(n):= inf t > τk−1 |St − Sτ (n) | = 2−n .


k−1Then we introduce the discretized price process(n)

Sk:= Sτ (n) ,k = 0, 1, . . .

kEyal Neuman | Imperial College London

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Optimal portfolio liquidation in target zone models

Temporary Impact Costs Caused by {Xtξ }t≥0Then we introduce the discretized price process(n)

Sk:= Sτ (n) ,k = 0, 1, . . .

kThe local time of S (n) in c is usually defined as(n)`k := 2−nkXi=0Eyal Neuman | Imperial College London

1

(n)

{Si

=c}

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Optimal portfolio liquidation in target zone models

Then we introduce the discretized price process(n)

Sk:= Sτ (n) ,k = 0, 1, . . .

kThe local time of S (n) in c is usually defined as(n)`k := 2−n

k
X
i=0

1

(n)

{Si

.
=c}

A generalisation of a result in [Le Gall 1994] gives us(n)

Sb22n tc → St

and

(n)

`b22n tc → Lt , uniformly in t, P − a.s.

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Optimal portfolio liquidation in target zone models

Temporary Impact Costs Caused by {Xtξ }t≥0
Define(n)ξk:= ξτ (n)
k

and

ξ,(n)

XN

:= x0 +NX(n) (n)(n)ξk (`k − `k−1 )k=0

1. ξ (n) is the speed, relative to the local time `(n) , at which shares are
sold or purchased.
ξ,(n)

2. XN
is the inventory of the investor at the N th time step of thediscrete-time approximation.
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Optimal portfolio liquidation in target zone models

Temporary Impact Costs Caused by {Xtξ }t≥01. Each executed order generates a temporary price impact,which is given by a function g of the trading speed (as in theframework of the Almgren-Chriss model)

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Optimal portfolio liquidation in target zone models

Temporary Impact Costs Caused by {Xtξ }t≥01. Each executed order generates a temporary price impact,which is given by a function g of the trading speed (as in theframework of the Almgren-Chriss model)
2. Here we take g(x) = sign(x)|x|p−1 for some p > 1,
which is consistent with [Almgren 03, Almgren, Hauptman, Li 05, Gatheral 10].

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Optimal portfolio liquidation in target zone models

Temporary Impact Costs Caused by {Xtξ }t≥01. Each executed order generates a temporary price impact,which is given by a function g of the trading speed (as in theframework of the Almgren-Chriss model)
2. Here we take g(x) = sign(x)|x|p−1 for some p > 1, whichis consistent with [Almgren 03, Almgren, Hauptman, Li 05,
Gatheral 10].
(n)

3. The speed of the k th order is ξk and the number of(n) (n)(n)shares executed by that order is ξk (`k − `k−1 ).

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Optimal portfolio liquidation in target zone models

Temporary Impact Costs Caused by {Xtξ }t≥02. Here we take g(x) = sign(x)|x|p−1 for some p > 1.
(n)

3. Since the speed of the k th order is ξk and the number of(n) (n)(n)shares executed by that order is ξk (`k − `k−1 ).
4. It follows that the total transaction costs incurred by
the first N orders are equal to
N
X

(n)

(n)

(n)

|ξk |p (`k − `k−1 ).

k=0

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Optimal portfolio liquidation in target zone models

Temporary Impact Costs Caused by {Xtξ }t≥0
The following result now provides the financial interpretation of the cost
minimization problem. We assume here that ξt has a Pz -a.s continuousversion.
Proposition
Under the above assumptions, we have that Pz -a.s. for each t > 0,b22n tc
ξ,(n)Xb22n tc −→ Xtξ

and

X

(n)

(n)

(n)

|ξk |p (`k −`k−1 ) −→k=0

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Z t|ξs |p L(ds).

0

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Optimal portfolio liquidation in target zone models

Financial Motivation: The Control Problem
Recall that we wish to minimize of the cost functional for some p ≥ 2,

Z T

C([0, T ]) = Ez

|ξt |p L(dt) +

0

Z T
0

φ(St )|Xtξ |p dt + %(ST )|XTξ |p• The term 0T |ξt |p L(dt) can be interpreted as a cost term that arises
from the temporary price impact generated by executing the strategy
Xξ.
R

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Optimal portfolio liquidation in target zone models

What is a superprocesses ?
N (t)1. Define {Sti }i=1 - a collection of critical branching diffusion particles
that live in R.
2. N (t) is the number of the particles in the system at time t.
3. We assume that between branching events the particles follow independent diffusion paths which are independent.

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Optimal portfolio liquidation in target zone models

What is a superprocesses ?
N (t)1. Define {Sti }i=1 - a collection of critical branching diffusion particles
that live in R.
2. N (t) is the number of the particles in the system at time t.
3. We assume that between branching events the particles follow independent diffusion paths which are independent.
4. Critical branching means that each particle splits into two or dieswith equal probability (independently of other particles).

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Optimal portfolio liquidation in target zone models

What is a superprocesses ?
3. We assume that between branching events the particles follow independent diffusion paths which are independent.
4. Critical branching means that each particle splits into two or dieswith equal probability (independently of other particles).
5. We assume that the times between branching are independently distributed exponential random variables with mean
1/m.
6. In what follows m is "large" (fast branching), N (0) ∼ m.

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Optimal portfolio liquidation in target zone models

What is a superprocesses ?
1. We define the following measure valued process
N (t)

(m)
Yt (A) =1 X
δ (i) (A), A ⊂ R.
m i=1 St

Here δx is the delta measure centred at x.
2. Suppose that {Y0m }m≥1 converges weakly to µ, as m → ∞.
3. In the appropriate topology, {Ytm }t≥0 converges weakly to a limiting
process {Yt }t≥0 , which is called superporcess.
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Optimal portfolio liquidation in target zone models

What is a Catalytic-Superprocesses ?
(with a single point catalyst at c)
We assume that the probability that a particle survives between[r, t] and dies between [t, t + dt] is given bye−L(r,t) dL(t),where {L(t)}t≥0 is the local time that the particle spends at the
point c between [0, t].

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Optimal portfolio liquidation in target zone models

Unique Solution to the Control Problem

1. Consider the catalytic superprocess Yt with a single point
catalyst at c.

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Optimal portfolio liquidation in target zone models

Unique Solution to the Control Problem
1. Consider the catalytic superprocess Yt with a single point
catalyst at c.
2. Lethu(t, z) := − log Eδz exp −Eyal Neuman | Imperial College London

Z t

i

hφ, Ys i ds − h%, Yt i

.

0

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Optimal portfolio liquidation in target zone models

Unique Solution to the Control Problem
1. Consider the catalytic superprocess Yt with a single point
catalyst at c.
2. Lethu(t, z) := − log Eδz exp −

Z t

i

hφ, Ys i ds − h%, Yt i.

0

3. Recall that X denote the class of all progressively measurable
R
control processes ξ for which 0T |ξt | dLt < ∞ Pz -a.s. for all
T > 0 and z ∈ R.
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Optimal portfolio liquidation in target zone models

Theorem (N. and Schied 2016)
Let β := 1/(p − 1) and

ξt∗ := −x0 exp −Z t
0

so that∗u(T − s, Ss )β dLs u(T − t, St )β

Xtξ = x0 exp −Z tu(T − s, Ss )β dLs .

0∗
Then ξ is the unique strategy in X minimizing the cost functional.

Moreover, the minimal cost is given by
C([0, T ]) = |x0 |p u(T, z).

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Optimal portfolio liquidation in target zone models

Current Research
1. The central bank point of view: for any given trader strategy
ξ ∈ X , the actual price process isS̃tξ = St + γ(Xtξ − x0 ).
What is the optimal strategy of the central bank which keeps {S̃tξ }≥0
above level c ?

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Optimal portfolio liquidation in target zone models

Current Research
1. The central bank point of view: for any given trader strategy ξ ∈ X ,the actual price process isS̃tξ = St + γ(Xtξ − x0 ).
What is the optimal strategy of the central bank which keeps {S̃tξ }≥0
above level c ?
2. Formulation of the trader-central bank system as a stochastic
game.

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Optimal portfolio liquidation in target zone models

Current Research
1. The central bank point of view: for any given trader strategy ξ ∈ X ,the actual price process isS̃tξ = St + γ(Xtξ − x0 ).
What is the optimal strategy of the central bank which keeps {S̃tξ }≥0
above level c ?
2. Formulation of the trader-central bank system as a stochastic game.
3. Is there an equilibrium between the central bank and thetrader’s optimal strategies ?
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Optimal portfolio liquidation in target zone models

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Optimal portfolio liquidation in target zone models

Connection between Control and Superprocesses
1. Super-Brownian motion {Yt }t≥0 satisfies
Eµ [e−hYt ,φi ] = e−hv(r,·),µi ,for every test function φ.

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Optimal portfolio liquidation in target zone models

Connection between Control and Superprocesses
1. Super-Brownian motion {Yt }t≥0 satisfies
Eµ [e−hYt ,φi ] = e−hv(r,·),µi ,for every test function φ.
2. The log-Laplace functional v satisfies∂v
1= ∆v − v 2 , v|t=0+ = φ.
∂t2Eyal Neuman | Imperial College London

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Optimal portfolio liquidation in target zone models

Connection Between Control and Superprocesses
In [Schied, 2013] the following value function was introduced

Z T

V (t, z, x0 ) := inf Et,zx()

2|ẋ(u)| du +

t

Z T

2



|x(u)| a(Wu )du .

t

1. Here W is a standard Brownian motion and a is some positive
measurable function.

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Optimal portfolio liquidation in target zone models

Connection Between Control and Superprocesses
In [Schied, 2013] the following value function was introduced

Z T

V (t, z, x0 ) := inf Et,zx()|ẋ(u)|2 du +

t

Z T



|x(u)|2 a(Wu )du .

t

1. Here W is a standard Brownian motion and a is some positive
measurable function.
2. The infimum is taken over the class of all absolutely continues
adapted strategies x() such that x(t) = x0 and x(T ) = 0.
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Optimal portfolio liquidation in target zone models

Connection Between Control and Superprocesses
In [Schied, 2013] the following value function was introduced

Z T

V (t, z, x0 ) := inf Et,zx()|ẋ(u)|2 du +

t

Z T



|x(u)|2 a(Wu )du .

t

The associated HJB equation is1
Vt (t, z, x0 ) + inf |ζ|2 + Vx0 (t, z, x0 )ζ + a(z)|x0 |2 + ∆V (t, z, x0 ) = 0.
ζ2Eyal Neuman | Imperial College London

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Optimal portfolio liquidation in target zone models

Connection Between Control and Superprocesses
In [Schied, 2013] the following value function was introduced

Z T

V (t, z, x0 ) := inf Et,zx()

2|ẋ(u)| du +

t

Z T

2



|x(u)| a(Wu )du .

t

The associated HJB equation is1
Vt (t, z, x0 ) + inf |ζ|2 + Vx0 (t, z, x0 )ζ + a(z)|x0 |2 + ∆V (t, z, x0 ) = 0,ζ2with V (T, z, x0 ) = 0 if x0 = 0 and V (T, z, x0 ) = ∞ otherwise.
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Optimal portfolio liquidation in target zone models

Connection Between Control and Superprocesses
The associated HJB equation is1
Vt (t, z, x0 )+inf ζ 2 +Vx0 (t, z, x0 )ζ +a(z)|x0 |2 + ∆V (t, z, x0 ) = 0,ζ2with V (T, z, x0 ) = 0 if x0 = 0 and V (T, z, x0 ) = ∞ otherwise.
For x0 > 0, assume that V (t, z, x0 ) = x20 v(t, z) for some function v.

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Optimal portfolio liquidation in target zone models

Connection between control problems and superprocesses
If we minimize over ζ we get that v formally stratifies:
 ∂v
1
2

 ∂t = − 2 ∆v + v − a,v(T, z) = +∞.

The Log-Laplace functional of SBM with branching rate 1 satisfies∂v
1= ∆v − v 2 , v|t=0+ = φ.
∂t2Eyal Neuman | Imperial College London

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Questions?